A bipartite consensus method for fuzzy multi-agent systems based on observers

Abstract

The present invention discloses a bipartite consistency method for a fuzzy multi-agent system based on an observer, which belongs to the technical field of bipartite consistency of a multi-agent system, and solves the technical problem of bipartite consistency of a nonlinear multi-agent system with a cooperative-competitive interactive relationship when the system state is difficult to measure directly. The technical scheme is as follows: it includes the following steps: considering a class of nonlinear multi-agent systems consisting of N nodes, establishing a T-S fuzzy multi-agent system model; then designing an observer system for the T-S fuzzy multi-agent system; then designing an observer-based controller, in which the symbolic diagram of the controller is different from the symbolic diagram of the observer; establishing a bipartite consistency condition for the fuzzy multi-agent system, obtaining the observer gain and the controller gain, and enabling the fuzzy multi-agent system to achieve bipartite consistency. The bipartite consistency method for a T-S fuzzy multi-agent system based on an observer proposed by the present invention is simple, convenient, and has wider applicability.

Description

Technical Field

[0001] The present invention relates to the technical field of bipartite consistency of multi-agent systems, and in particular to an observer-based fuzzy bipartite consistency method for multi-agent systems. Background Art

[0002] In recent years, the cooperative behavior of multi-agent systems has attracted great interest due to their wide applications in formation control and swarm control. Among them, as one of the key issues in the cooperative behavior of multi-agent systems, the consistency of multi-agent systems has received extensive attention. The network cooperation-competition interaction relationship is usually described by a signed graph with both positive and negative edges. In the past decade, people have studied the bipartite consistency of multi-agent systems with various dynamics in cooperative-competition interaction signed networks, such as linear dynamics and Lipschitz nonlinear dynamics systems.

[0003] As we all know, the Takagi-Sugeno (TS) fuzzy model is an effective tool for dealing with nonlinear problems in complex network systems. At present, there are few studies on the bipartite consensus of TS fuzzy multi-agent systems with cooperative-competitive interactive signed networks, such as: Fast fixed-time impulsive bipartite synchronization of TS fuzzy complex networks with signed graphs (DOI: 10.1016 / j.nahs.2022.101325), Bipartite consensus for Takagi-Sugeno fuzzy uncertain multi-agent systems with gain fluctuations (DOI: 10.1109 / TSIPN.2023.3239679). However, the only studies on the bipartite consensus of TS fuzzy multi-agent systems ignore a problem that the system state may not be directly measured in actual situations, and it is necessary to study it based on observers. At present, there is no report on the bipartite consistency study based on observer for TS fuzzy multi-agent systems with cooperative-competitive interactive symbolic networks. In this case, the present invention proposes a bipartite consistency method for TS fuzzy multi-agent systems based on observer. Summary of the invention

[0004] In order to solve the problem of bipartite consistency in nonlinear multi-agent systems with cooperative-competitive interactions when it is difficult to directly measure the system state, an observer-based TS fuzzy multi-agent system bipartite consistency method was proposed.

[0005] Technical solution: A bipartite consistency method for fuzzy multi-agent systems based on observers, including the following steps:

[0006] Step 1: Consider a nonlinear multi-agent system consisting of N nodes, model the multi-agent system, and establish a TS fuzzy multi-agent system model, as shown in the following formula (1):

[0007] Agent rule m: If ζ1(t) is Π m1 , ζ2(t) is Π m2 ,…,ζ p (t) is Π mp ,but

[0008]

[0009] Among them, d (t) and Π md are antecedent variables and fuzzy sets respectively, d=1,2,…,p, m=1,2,…,r, r is the number of fuzzy rules, is the state variable of node i, is the output measurement value of node i, u i (t) is the control input, the matrix and are all known constant real number matrices;

[0010] The global TS fuzzy model of the multi-agent system is shown in the following formula (2):

[0011]

[0012] where ζ=[ζ1,ζ2,...,ζ p ], is the normalized membership function, expressed as Satisfy h m (ζ)≥0 and but and Π md (ζ d ) is ζ d In Π md The degree of membership in ;

[0013] Step 2: Design the observer system of the TS fuzzy multi-agent system. The observer model is expressed by the following formula (3):

[0014] Observer rule m: If ζ1(t) is Π m1 , ζ2(t) is Π m2 ,…,ζ p (t) is Π mp ,but

[0015]

[0016] in is the estimated state variable of node i, is the output measurement error of node i, L m is the observer gain to be designed, m=1,2,...,r, This is the symbol diagram of the output measurement error. The adjacency matrix in ;

[0017] The global TS fuzzy model of the observer system is shown in the following formula (4):

[0018]

[0019] Step 3: Design an observer-based controller. The controller model is shown in the following formula (5):

[0020] Controller rule m: If ζ1(t) is ∏ m1 , ζ2(t) is Π m2 ,…,ζ p (t) is mp ,but

[0021]

[0022] Among them, K m is the controller gain to be designed, m=1,2,...,r, is a symbolic diagram of the estimated state The adjacency matrix in ;

[0023] The global TS fuzzy model of the controller is shown in the following formula (6):

[0024]

[0025] Step 4: Establish the bipartite consistency condition of the fuzzy multi-agent system (2) and obtain the observer gain L m and controller gain K m , so that the fuzzy multi-agent system (2) achieves bipartite consistency;

[0026] If there exists a matrix X>0,Y>0,Y1, and So that the following inequality holds:

[0027] Ψ mm <0, m=1,…,r (7)

[0028]

[0029] CY=Y1C (9)

[0030] in, in,

[0031] Get the observer gain L m and controller gain K m for,

[0032]

[0033] Then the fuzzy multi-agent system (2) can achieve bipartite consistency.

[0034] Furthermore, the step 2 of the observer-based fuzzy multi-agent system bipartite consensus method is described in This is the symbol diagram of the output measurement error. The adjacency matrix in , where the adjacency matrix It is defined as: when the output measurement error of the i-th agent and the output measurement error of the jth agent If there is a connection between them, then h ij =h ji ≠0, j≠i; otherwise, h ij =h ji =0, j≠i;

[0035] When h ij =h ji When >0, it means that there is a cooperative relationship between the i-th agent observer and the j-th agent observer;

[0036] When h ij =h ji When <0, it means that there is a competitive relationship between the i-th agent observer and the j-th agent observer.

[0037] Furthermore, the step 2 of the observer-based fuzzy multi-agent system bipartite consensus method is described in This is the symbol diagram of the output measurement error. The adjacency matrix in , where the symbolic graph Self-loops are not included.

[0038] Furthermore, the step 3 of the observer-based fuzzy multi-agent system bipartite consensus method is is a symbolic diagram of the estimated state The adjacency matrix in , where the adjacency matrix Defined as: when the estimated state of the i-th agent and the estimated state of the jth agent If there is a connection between ij =g ji ≠0, j≠i; otherwise, g ij =g ji =0, j≠i;

[0039] When ij =g ji When >0, it indicates that there is a cooperative relationship between the controller of agent i and the controller of agent j;

[0040] When ij =g ji <0, it means that there is a competitive relationship between the controller of agent i and the controller of agent j.

[0041] Furthermore, the step 3 of the observer-based fuzzy multi-agent system bipartite consensus method is is a symbolic diagram of the estimated state The adjacency matrix in , where the symbolic graph Self-loops are not included.

[0042] Furthermore, the symbolic graph described in step 2 of the observer-based fuzzy multi-agent system bipartite consensus method and the symbol diagram described in step 3 are interconnected.

[0043] Furthermore, the symbolic graph described in step 2 of the observer-based fuzzy multi-agent system bipartite consensus method and the symbol diagram described in step 3 All are structurally balanced and have the same bipartite non-empty node subsets and Right now Satisfies: When agent i and agent j belong to the same subset, then h ij ≥0,g ij ≥0; otherwise h ij ≤0,g ij ≤0.

[0044] Compared with the prior art, the present invention has the following beneficial effects:

[0045] (1) The present invention comprehensively considers the joint influence of system nonlinearity and network cooperation-competition interaction, adopts TS fuzzy model to deal with system nonlinearity problem, uses symbolic graph to describe network cooperation-competition interaction, and combines the actual situation that it is often difficult to directly measure the system state to design an observer-based controller to solve the bipartite consistency problem of nonlinear multi-agent system with cooperation-competition interaction;

[0046] (2) The present invention proposes an observer-based binary consistency control method for fuzzy multi-agent systems. This method is more general than some existing methods, such as feedback control methods based on state information and control methods based on distributed observers. This is because existing control methods require that the network topology of the observer must be the same as the network topology of the controller, while the present invention allows the two to be different, which makes the application range of the present invention wider and the design simpler. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Figure 1 This is a schematic diagram of the bipartite consistency method of a fuzzy multi-agent system based on an observer of the present invention.

[0048] Figure 2 is the state x of the fuzzy multi-agent system in the numerical example of the present invention i (t) Trajectory diagram; where (a) is the state x of the fuzzy multi-agent system i1 (t) is the trajectory diagram; (b) is the state x of the fuzzy multi-agent system i2 Trajectory diagram of (t), i=1,…,5.

[0049] Figure 3 is the state estimation error e of the fuzzy multi-agent system in the numerical example of the present invention i (t) trajectory diagram; where (a) is the state estimation error e of the fuzzy multi-agent system i1 (t) is the trajectory diagram; (b) is the state estimation error e of the fuzzy multi-agent system i2 Trajectory diagram of (t), i=1,…,5.

[0050] Figure 4 is the state estimation bipartite consistency error signal of the fuzzy multi-agent system in the numerical example of the present invention Trajectory diagram; where (a) is the state estimation bisection consistency error signal (b) is the trajectory diagram of the state estimation bisection consistency error signal Trajectory diagram of , i=1,…,5. DETAILED DESCRIPTION

[0051] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. Of course, the specific embodiments described here are only used to explain the present invention and are not used to limit the present invention.

[0052] As attached Figure 1 As shown, the present embodiment proposes an observer-based bipartition consistency method for a multi-agent system. Considering a nonlinear multi-agent system consisting of N nodes, the multi-agent system is modeled and a TS fuzzy multi-agent system model is established, as shown in the following formula (11):

[0053] Agent rule m: If ζ1(t) is Π m1 , ζ2(t) is Π m2 ,…,ζ p (t) is Π mp ,but

[0054]

[0055] Among them, d (t) and Π md are antecedent variables and fuzzy sets respectively, d=1,2,…,p, m=1,2,…,r, r is the number of fuzzy rules. is the state variable of node i, is the output measurement value of node i, u i (t) is the control input. The matrix and are all known constant real number matrices.

[0056] According to the fuzzy reasoning method, the global TS fuzzy model of the multi-agent system is shown in the following formula (12):

[0057]

[0058] where ζ=[ζ1,ζ2,...,ζ p ], is the normalized membership function, expressed as Satisfy h m (ζ)≥0 and but and Π md (ζ d ) is ζ d In Π md The degree of membership in .

[0059] Since it is difficult to estimate the system state in practical situations, the observer system of the multi-agent system (12) is proposed as follows:

[0060] Observer rule m: If ζ1(t) is Π m1 , ζ2(t) is Π m2 ,…,ζ p (t) is Π mp ,but

[0061]

[0062] in is the estimated state variable of node i, is the output measurement error of node i, L m is the observer gain to be designed, m=1,2,...,r, This is the symbol diagram of the output measurement error. The adjacency matrix in is defined as: when the output measurement error of the i-th agent and the output measurement error of the jth agent If there is a connection between them, then h ij =h ji ≠0, j≠i; otherwise, h ij =h ji = 0, j ≠ i. In particular, when h ij =h ji >0, it indicates that there is a cooperative relationship between the i-th agent observer and the j-th agent observer; when h ij =h ji When <0, it means that there is a competitive relationship between the i-th agent observer and the j-th agent observer.

[0063] The global TS fuzzy model of the observer system is shown in formula (14):

[0064]

[0065] In order to achieve binary synchronization, an observer-based controller is designed. The controller model is as follows:

[0066] Controller rule m: If ζ1(t) is Π m1 , ζ2(t) is Π m2 ,…,ζ p (t) is Π mp ,but

[0067]

[0068] Among them, K m is the controller gain to be designed, m=1,2,...,r, is a symbolic diagram of the estimated state The adjacency matrix in is defined as: when the estimated state of the i-th agent and the estimated state of the jth agent If there is a connection between ij =g ji ≠0, j≠i; otherwise, g ij =g ji = 0, j ≠ i. In particular, when g ij =g ji When g > 0, it means that the controller of agent i and the controller of agent j are in a cooperative relationship; when g ij =g ji When <0, it means that there is a competitive relationship between the controller of agent i and the controller of agent j.

[0069] The global TS fuzzy model of the controller is shown in the following formula (16):

[0070]

[0071] Command symbol diagram and middle, AND Sign Diagram and Self-loops are not included.

[0072] Symbol diagram The Laplacian matrix of is defined as: in Symbol diagram The Laplacian matrix of is defined as: in Let Γ = diag{γ1,...,γ N}, where: When γ i =1; when When γ i =-1.

[0073] Assumption 1. Symbolic graph and are interconnected.

[0074] Assumption 2. Symbolic graph and are structurally balanced and have the same bipartite non-empty subset of nodes and Right now Satisfies: When agent i and agent j belong to the same subset, then h ij ≥0,g ij ≥0; otherwise h ij ≤0,g ij ≤0.

[0075] Definition 1. Bipartite consistency can be achieved if the following conditions are met:

[0076]

[0077] Among them, when When γ i =1, otherwise, γ i =-1.

[0078] The state estimation error signal is defined as The error system of the observer system (14) is obtained as follows,

[0079]

[0080] Considering systems (14) and (16) simultaneously, we obtain

[0081]

[0082] Let z i (t) = γ i x i (t), From systems (18) and (19), we can obtain

[0083]

[0084]

[0085] in,

[0086] Define the bipartite consistency error signal of the state estimation as The bipartite consistency error signal of the state estimation error is Then from systems (21) and (20) we can obtain

[0087]

[0088]

[0089] make in

[0090] in

[0091] The enhanced system is

[0092]

[0093] in

[0094] Lemma 1. Assume that the symbolic graph Contains a spanning tree and is structurally balanced. Laplacian matrix has one zero eigenvalue, and the other N-1 eigenvalues ​​are all located in the right half plane.

[0095] Theorem 1. Under the assumptions 1 and 2, if there exists a matrix X>0, Y>0, Y1, and So that the following inequality holds:

[0096] Ψ mm <0,m=1,...,r (25)

[0097]

[0098] CY=Y1C (27)where,

[0099] At the same time, the observer gain L m and controller gain K m for,

[0100]

[0101] Then the fuzzy multi-agent system (12) can achieve bipartite consistency.

[0102] Numerical Examples

[0103] Consider a nonlinear multi-agent system with 5 nodes, where each agent node is a tunnel diode circuit modeled by a TS fuzzy system with parameters: i1 (t)∈[-3,3], C=[1 0].

[0104] The 5 agent nodes are divided into two subsets, V1 = {1, 2, 3} and V2 = {4, 5}, symbolic diagram and Assumptions 1 and 2 are satisfied. The Laplacian matrix and They are

[0105] According to Theorem 1, the gain matrix is ​​as follows:

[0106] The controller gains are K1 = [-0.1202 -0.1823], K2 = [-0.1379 -0.3251];

[0107] The observer gain is

[0108] The membership function in the simulation is taken as Attached Figure 2 , Attachment Figure 3 and attached Figure 4 The states x of the fuzzy multi-agent system are shown respectively. i (t) Trajectory and state estimation error e i (t) Trajectory and state estimation bisection consistency error signal As can be seen from the attached figure, the designed observer-based controller achieves the bipartite consistency of the TS fuzzy multi-agent system, and the designed observer also effectively estimates the unmeasurable system state.

[0109] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims

1. A bipartite consensus method for fuzzy multi-agent systems based on an observer, characterized in that: The steps include: Step 1: Consider a nonlinear multi-agent system consisting of N nodes, model the multi-agent system, and establish a TS fuzzy multi-agent system model, as shown in the following formula (1): Agent rule m: If ζ1(t) is Π m1 , ζ2(t) is m2 ,…,ζ p (t) is П mp ,but Among them, d (t) and ∏ md are antecedent variables and fuzzy sets respectively, d=1,2,...,p, m=1,2,...,r, r is the number of fuzzy rules, is the state variable of node i, is the output measurement value of node i, u i (t) is the control input, the matrix and are all known constant real number matrices; The global TS fuzzy model of the multi-agent system is shown in the following formula (2): where ζ=[ζ1,ζ2,...,ζ p ], is the normalized membership function, expressed as Satisfy h m (ζ)≥0 and but and П md (ζ d ) is ζ d In md The degree of membership in ; Step 2: Design the observer system of the TS fuzzy multi-agent system. The observer model is expressed by the following formula (3): Observer rule m: If ζ1(t) is ∏ m1 , ζ2(t) is Π m2 ,…,ζ p (t) is Π mp ,but in is the estimated state variable of node i, is the output measurement error of node i, L m is the observer gain to be designed, m=1,2,...,r, This is the symbol diagram of the output measurement error. The adjacency matrix in ; The global TS fuzzy model of the observer system is shown in the following formula (4): Step 3: Design an observer-based controller. The controller model is shown in the following formula (5): Controller rule m: If ζ1(t) is ∏ m1 , ζ2(t) is Π m2 ,…,ζ p (t) is Π mp ,but Among them, K m is the controller gain to be designed, m=1,2,...,r, is a symbolic diagram of the estimated state The adjacency matrix in ; The global TS fuzzy model of the controller is shown in the following formula (6): Step 4: Establish the bipartite consistency condition of the fuzzy multi-agent system (2) and obtain the observer gain L m and controller gain K m , so that the fuzzy multi-agent system (2) achieves bipartite consistency; If there exists a matrix X>0,Y>0,Y1, and So that the following inequality holds: P mm <0, m=1,...,r (7) CY=Y1C (9) in, in, Get the observer gain L m and controller gain K m for, Then the fuzzy multi-agent system (2) can achieve bipartite consistency.

2. The observer-based fuzzy multi-agent system bipartite consensus method according to claim 1, characterized in that: In the step 2, the This is the symbol diagram of the output measurement error. The adjacency matrix in , where the adjacency matrix It is defined as: when the output measurement error of the i-th agent and the output measurement error of the jth agent If there is a connection between them, then h ij =h ji ≠0, j≠i; otherwise, h ij =h ji =0, j≠i; When h ij =h ji When >0, it means that there is a cooperative relationship between the i-th agent observer and the j-th agent observer; When h ij =h ji When <0, it means that there is a competitive relationship between the i-th agent observer and the j-th agent observer.

3. The observer-based fuzzy multi-agent system bipartite consensus method according to claim 2, characterized in that: The step 2 described This is the symbol diagram of the output measurement error. The adjacency matrix in , where the symbolic graph Self-loops are not included.

4. The observer-based fuzzy multi-agent system bipartite consensus method according to claim 3, characterized in that: In the step 3, the is a symbolic diagram of the estimated state The adjacency matrix in , where the adjacency matrix Defined as: when the estimated state of the i-th agent and the estimated state of the jth agent If there is a connection between ij =g ji ≠0, j≠i; otherwise, g ij =g ji =0, j≠i; When ij =g ji When >0, it means that there is a cooperative relationship between the controller of agent i and the controller of agent j; When ij =g ji When <0, it means that there is a competitive relationship between the controller of agent i and the controller of agent j.

5. The observer-based fuzzy multi-agent system bipartite consensus method according to claim 4, characterized in that: In the step 3, the is a symbolic diagram of the estimated state The adjacency matrix in , where the symbolic graph Self-loops are not included.

6. The observer-based fuzzy multi-agent system bipartite consensus method according to claim 5, characterized in that: The symbol diagram And symbol diagram It refers to the symbol diagram and symbol diagram are interconnected.

7. The observer-based fuzzy multi-agent system bipartite consensus method according to claim 5, characterized in that: The symbol diagram And symbol diagram It refers to the symbol diagram and All are structurally balanced and have the same bipartite non-empty node subsets and Right now Satisfies: When agent i and agent j belong to the same subset, then h ij ≥0,g ij ≥0; otherwise h ij ≤0,g ij ≤0.

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